Applications of Mathematics

, Volume 64, Issue 2, pp 253–277 | Cite as

On parameter estimation in an in vitro compartmental model for drug-induced enzyme production in pharmacotherapy

  • Jurjen Duintjer TebbensEmail author
  • Ctirad MatonohaEmail author
  • Andreas MatthiosEmail author
  • Štěpán PapáčekEmail author


A pharmacodynamic model introduced earlier in the literature for in silico prediction of rifampicin-induced CYP3A4 enzyme production is described and some aspects of the involved curve-fitting based parameter estimation are discussed. Validation with our own laboratory data shows that the quality of the fit is particularly sensitive with respect to an unknown parameter representing the concentration of the nuclear receptor PXR (pregnane X receptor). A detailed analysis of the influence of that parameter on the solution of the model’s system of ordinary differential equations is given and it is pointed out that some ingredients of the analysis might be useful for more general pharmacodynamic models. Numerical experiments are presented to illustrate the performance of related parameter estimation procedures based on least-squares minimization.


pharmacotherapy pharmacodynamic modelling constrained optimization parameter estimation 

MSC 2010

92C45 34A34 65F60 65K10 


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We would like to thank Prof. Petr Pávek for the laboratory experiments described in this manuscript; they were performed under his supervision and in his laboratory at the Faculty of Pharmacy of Charles university in Hradec Králové.


  1. [1]
    D. Z. D’Argenio, A. Schumitzky, X. Wang: ADAPT 5 User’s Guide: Pharmacokinetic/Pharmacodynamic Systems Analysis Software. Biomedical Simulations Resource, Los Angeles, 2009, Available at Scholar
  2. [2]
    S. Dhillon, A. Kostrzewski, (eds.): Clinical Pharmacokinetics. Pharmaceutical Press, London, 2006.Google Scholar
  3. [3]
    J. Duintjer Tebbens, M. Azar, E. Friedmann, M. Lanzendörfer, P. Pávek: Mathematical models in the description of pregnane X receptor (PXR)-regulated cytochrome P450 enzyme induction. Int. J. Mol. Sci. 19(2018), 1785.CrossRefGoogle Scholar
  4. [4]
    A. Funahashi, M. Morohashi, H. Kitano, N. Tanimura: CellDesigner: a process diagram editor for gene-regulatory and biochemical networks. BIOSILICO 1(2003), 159–162.CrossRefGoogle Scholar
  5. [5]
    The GNU Fortran compiler. Available at
  6. [6]
    A. C. Hindmarsh: Large ordinary differential equation systems and software. Control Systems Magazine 2 (1982), 24–30.CrossRefGoogle Scholar
  7. [7]
    A. C. Hindmarsh: ODEPACK, a systematized collection of ODE solvers. Scientific Computing 1982 (R. S. Stepleman et al., eds.). IMACS Transactions on Scientific Computation I, North-Holland Publishing, Amsterdam, 1983, pp. 55–64.Google Scholar
  8. [8]
    H. M. Jones, K. Rowland-Yeo: Basic Concepts in Physiologically Based Pharmacokinetic Modeling in Drug Discovery and Development. CPT: Pharmacometrics & Systems Pharmacology 2 (2013), Article ID e63, 12 pages.Google Scholar
  9. [9]
    N. S. Luke, M. J. DeVito, I. Shah, H. A. El-Masri: Development of a quantitative model of pregnane X receptor (PXR) mediated xenobiotic metabolizing enzyme induction. Bull. Math. Biol. 72 (2010), 1799–1819.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    L. Lukšan, M. Tůma, C. Matonoha, J. Vlček, N. Ramešová, M. Šiška, J. Hartman: UFO 2017. Interactive System for Universal Functional Optimization. Technical Report V-1252, Institute for Computer Science CAS, Praha, 2017. Available at Scholar
  11. [11]
    D. J. Lunn, N. Best, A. Thomas, J. Wakefield, D. J. Spiegelhalter: Bayesian analysis of population PK/PD models: General concepts and software. J. Pharmacokinetics Pharmacodynamics 29 (2002), 271–307.CrossRefGoogle Scholar
  12. [12]
    MATLAB. Mathworks, Inc., 2018. Available at
  13. [13]
    C. Moler, C. Van Loan: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45 (2003), 3–49.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. A. Nelder, R. Mead: A simplex method for function minimization. Computer J. 4 (1965), 308–313.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    NONMEM 7.3. ICON, Inc., 1990–2016. Available at
  16. [16]
    L. Petzold: Automatic selection of methods for solving stiff and nonstiff systems of ordinary diferential equations. SIAM J. Sci. Stat. Comput. 4 (1983), 136–148.CrossRefzbMATHGoogle Scholar
  17. [17]
    L. Shargel, A. B. C. Yu: Applied Biopharmaceutics & Pharmacokinetics. McGraw-Hill Education, New York, 2016.Google Scholar
  18. [18]
    Simcyp simulator. Certara, 2012. Available at
  19. [19]
    J. W. Spruill, W. E. Wade, T. J. DiPiro, A. R. Blouin, M. J. Pruemer: Concepts in Clinical Pharmacokinetics. American Society of Health-System Pharmacists, Bethesda, 2014.Google Scholar
  20. [20]
    P. Zhao, M. Rowland, S.-M. Huang: Best practice in the use of physiologically based pharmacokinetic modeling and simulation to address clinical pharmacology regulatory questions. Clinical Pharmacology & Therapeutics 92 (2012), 17–20.CrossRefGoogle Scholar
  21. [21]
    Z. Zheng, P. S. Stewart: Penetration of rifampin through staphylococcus epidermidis biofilms. Antimicrob. Agents Chemoter 46 (2002), 900–903.CrossRefGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Cz 2019

Authors and Affiliations

  1. 1.Faculty of Pharmacy in Hradec KrálovéCharles UniversityHradec KrálovéCzech Republic
  2. 2.Institute of Computer ScienceCzech Academy of SciencesPraha 8Czech Republic
  3. 3.Institute of Complex Systems, South Bohemian Research Center of Aquaculture and Biodiversity of Hydrocenoses, Faculty of Fisheries and Protection of WatersUniversity of South Bohemia in České BudějoviceNové HradyCzech Republic

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